Shrewsbury, under caretaker manager Mike Jackson for the second time, had not scored a goal at Greenhous Meadow in more than 10 hours of football until Eaves rifled a 15-yard shot to the top corner of the net to peg back the Bees. The Londoners had led through Marcello Trotta's 68th-minute goal.
The Shrews competed well in a goalless first half as they proved the equals of the Bees.
Eaves fired wide of the near post early on before Bahrudin Atajic, a striker on loan from Celtic, volleyed narrowly over from Paul Parry's left-wing cross.
Brentford started to ask more questions as the contest developed and twice the dangerous Trotta was narrowly off target.
Eaves thought he had scored early in the second half only for his header to be nodded off the line by Adam Forshaw.
Brentford, now 18 games unbeaten, broke the deadlock in the 68th minute. Alan Judge fell under a challenge from Ryan Woods with referee Mark Heywood pointing straight to the penalty spot.
Chris Weale saved Trotta's spot-kick only for the rebound to go straight back to the Italian to score his 11th goal of the season.
Shrewsbury were determined to end a run of six straight home defeats and thanks to Eaves, on loan from Bolton, they did just that.